Information-Based Validity Functionals for Mixture Analysis

Model identification is an essential but often neglected component of sound statistical inference. In mixture analysis, data are assumed to be sampled from a distribution with density $$ {f_{\theta }}(x) = \sum\limits_{{i = 1}}^k {{\pi_i}f\left( {x,\;{\theta_i}} \right),} $$ , where f(x, θ i ) is a member of a parametric family of densities. In this context, model identification can be thought of as obtaining estimates for the unknown parameters in the model. In the most general situation, the unknown parameters are k, the number of components in the model, π 1,…, π k, the mixing proportions, and θ 1 ,…, θ k , the parameters of the individual components. Estimating k, the number of components, is the most problematic aspect of the estimation problem. If k is known, then the other parameters can be estimated in a variety of ways, in particular, by maximum likelihood estimation. The typical approach to estimating k is to obtain estimates for the other parameters for a reasonable range of k’s and choose from among those models using a validity functional.